236 



ANIMAL MECHANICS. 



mined, when the line ab is carried 

 into the position of the line a'V '. 



the other point, then we can determine not only the axis upon which the 

 rotation took place, but also the position assumed by any other point resultant 

 from the same displacement. In Fig. 123 let the points a and b be points 

 upon the surface of a sphere with a large radius, and the points a and b' the 

 position which they occupy subsequent to a given movement. 



Join a and a, b and b' by arcs of a great circle, and bisect each of these 



arcs. From the point of bisection draw 

 / lines perpendicular to aa' and bb', meeting 

 at the point o. The point o will have 

 remained stationary during the rotation, 

 and so will C, the centre of rotation, as well 

 as every point on the line joining them. 

 The line passing from the centre through o 

 will therefore be the axis of rotation 

 required. For join oa, ob, and oa and ob', 

 then the triangles oab and o'a'b' are equal 

 and similar, hence the displacement of 

 o is zero. The angle of rotation will be 

 aoa. 



To find the position assumed by any 

 other point p, during the displacement of 

 the points ab to a'b', perform the following 

 ball-and-socket joint is deter- construction (see Fig. 124). 



Join pa and pb and ab to make the 

 triangle pab. Join a'b', and upon a'b', the 

 final position assumed by the line ab, erect 

 a triangle a'p'b' equal and similar to alp. The point p' will be the position 

 assumed by p at the end of the rotation. 



Compounding axes of rotation. If rotation occurs around two axes, 

 one after the other, the same result may lie obtained by a single 

 rotation around a definite 

 single axis. To find this 

 single axis, adopt the follow- 

 ing construction. In order ? 

 that the line of the axis may ^ 

 be made to represent the 

 direction of rotation, select 

 that half of each axis which, 

 if it runs through the length 

 of the body of an imaginary 

 observer, with his feet upon 

 the centre of rotation, will 

 place him in such a position 

 that the direction of rotation 

 will correspond to the direc- 

 tion of rotation of the hands 

 of a watch. 



Let the points O and O' represent the points at which these axes cut the 

 surface of the sphere ; join O and O'. If the rotation at O be equal to B, 



r\ 



measure off the angle O'O# equal to - and in a direction opposite to the rotation 



around O. If the rotation around O' be equal to 6', measure off the angle 



(9' 



OO'.'' equal to _ and passing in the same direction. The position of the 

 2 



FIG. 124. Shows the method of finding the posi- 

 tion assumed by the pointy, when the line ab 

 is rotated into the position a'b'. 



